Abstract

The use of co-occurrence data is common in various domains. Co-occurrence data often needs to be normalized to correct for the size effect. To this end, van Eck and Waltman (2009) recommend a probabilistic measure known as the association strength. However, this formula, based on combinations with repetition, implicitly assumes that observations from the same entity can co-occur even though in the intended usage of the measure these self-co-occurrences are nonexistent. A more accurate measure based on combinations without repetition is introduced here and compared to the original formula in mathematical derivations, simulations, and patent data, which shows that the original formula overestimates the relation between a pair and that some pairs are more overestimated than others. The new measure is available in the EconGeo package for R maintained by Balland (2016).

PEER REVIEW

1. INTRODUCTION

The use of co-occurrence data is popular in numerous scientific domains, such as scientometrics (e.g., Leydesdorff & Vaughan, 2006; van Eck & Waltman, 2009), computational linguistics (e.g., Schutze, 1998), community ecology (e.g., Peres-Neto, 2004), development economics (e.g., Hidalgo, Kilinger et al., 2007), molecular biology (e.g., Maslov & Sneppen, 2002) and evolutionary economic geography (e.g., Boschma, Balland, & Kogler, 2015). Its use is widespread and closely related to the popularity of network analysis across disciplines.

Co-occurrence data is used to infer the relation (referred to as relatedness here, following Hidalgo et al. [2007]), between entities, which can be species of fish, authors, or technological classes, by observing how each of these co-occur with others in places such as streams, articles, or patents. However, the total number of co-occurrences between a pair of entities cannot be used straightforwardly to reflect the relatedness between them because entities with more observations are more likely to co-occur than entities with fewer observations. To correct for this size effect, a normalization measure is applied to the data1. van Eck and Waltman (2009) review the most popular normalization measures and make a convincing case for the use of a probability-based measure known as the association strength. This measure is based on dividing the observed number of co-occurrences over the expected numbers of co-occurrences when assuming observations are randomly distributed over co-occurrences2.

In this paper, it is shown that the probability formula for the association strength, as proposed by van Eck and Waltman (2009), is not optimized to calculate the expected number of co-occurrences. The formula of van Eck and Waltman (2009) is proportional to probability calculations based on combinations with repetition, which means that when estimating the probability that two entities co-occur, the first observation drawn is assumed to be available for drawing again when drawing the second observation. However, in the use of co-occurrence data the co-occurrence of observations from the same entity is disregarded3. Authors, for example, do not coauthor papers with themselves (Leydesdorff & Vaughan, 2006). Therefore, van Eck and Waltman (2009) suggest setting these self co-occurrences to missing values4. This makes the possibility of drawing the same observation or any other observation from the same entity impossible in the second draw once an observation from this entity has been drawn in the first draw.

Therefore, an improved formula for the association strength is introduced using a probability measure based on combinations without repetition but with a noticeable change. In combinations with repetition, one cannot draw an observation in the second draw if it has been drawn in the first draw. In this setting, none of the observations belonging to the same entity as the first observation can be drawn in the second draw. Furthermore, two refinements are proposed in this paper regarding the inputs to the formula, which in the current definition do not properly take into account how the number of observed co-occurrences is calculated.

The improved formula is compared to the original formula in a theoretical setting, a number of simulations, and a real-world application using patent data. It is shown that, first, the original formula overestimates the relatedness between a pair of entities when this pair has at least one co-occurrence. This indicates that the original formula can wrongly identify two entities as related when in fact they are not; and, second, the original formula overestimates the relatedness between some pairs more than others. This indicates that the overestimation is not proportional and that the differences between the relatedness values for each pair are also distorted.

In the theoretical analysis, the improved formula is subtracted from the original formula to obtain a formula for the difference. By considering the domain of each variable, it is shown that the original formula underestimates the number of expected occurrences in all cases and therefore overestimates the relationship between two entities when there is at least one observed co-occurrence. Continuing the theoretical exploration, the first-order partial derivatives of the difference with respect to each variable is taken, which shows that the overestimation is not equal across all possible types of co-occurrence matrices.

Just taking the partial derivatives is not sufficient to show the size of the difference for each case, as the values of the variables are interconnected in ways that do not allow for analytical solving. Therefore, simulations are run in which four different exemplary cases are taken to the extreme to demonstrate the effect on the difference. The simulations show that the overestimation by the original formula can be close to 0% but also close to 100% of the relatedness value given by the improved formula, depending on the specificities of the co-occurrence matrix.

To measure to what extent these theoretical simulations are representative of real-world applications of research on co-occurrence data, a number of patent samples, containing data on the technology classes per document, is treated to compare the results of both formulas. In these samples, the overestimation of relatedness values for individual pairs varies between close to 0% to up to 3.234% of the value given by the improved formula, and therefore does not attain the most extreme values obtained in the simulation. Nonetheless, it clearly confirms that some pairs are more overestimated than others. The results also show that some pairs are misidentified as being related by the original formula, but that this is only the case for a rather small share of the pairs, up to about 0.29% of the number of pairs identified by the original formula.

Therefore, it is advisable to use the improved formula when working with co-occurrence data where self-co-occurrences are nonexistent or irrelevant. The reformulation of the probability measure does not in any way alter the conclusion by van Eck and Waltman (2009) that probability-based measures outperform so-called set-theoretic measures in normalizing co-occurrence data. The improved measure, including the recommended method of implementation, is available in the EconGeo package for R maintained by Balland (2016).

This paper is organized as follows: Section 2 gives a short overview of the use of co-occurrence data and the association strength; Section 3 discusses the refinements; Sections 4 to 6 explore the overestimation by the original formula respectively in a theoretical setting, simulations, and a real-world example using patent data; and Section 7 concludes.

2. NORMALIZING CO-OCCURRENCE DATA THROUGH PROBABILISTIC SIMILARITY MEASURES

Co-occurrence data is generally derived from a binary occurrence matrix O of some order m × n. The rows of O correspond to the places in which the observations occur and the columns to the entities to which they belong5. There is a large variety of what these places and entities can be6. The example in Matrix 1 shows three patents that contain a reference to, respectively, only class c, class c and class d, and all classes a to d.

Matrix 1

ClassaClassbClasscClassdPatent10010Patent20011Patent31111

By multiplying the transpose of O by O itself the co-occurrence matrix C is obtained7, in which both the rows and the columns represent the entities and the matrix gives how often they co-occur with the other.

In the case of our example, this would yield the co-occurrence matrix C given in Matrix 2. Where class a co-occurs once with b, c, and d; class b co-occurs once with a, c, and d; class c co-occurs once with a and b, and twice with d; and class d co-occurs once with a and b, and twice with c.

The diagonal is set to zero as the reference to a certain class does not entail a co-occurrence between that class and itself in the line of research for which the formula is intended. Ahlgren et al. (2003), Leydesdorff and Vaughan (2006) and van Eck and Waltman (2009) suggest setting the diagonal to missing values. This leads to the same results. However, it is advisable to use zeros, because missing values often results in errors when using statistical software8. Setting the diagonal to zero has important implications down the line.

Matrix 2

ClassaClassbClasscClassdClassa0111Classb1011Classc1102Classd1120

In many applications of co-occurrence data, such as the concept of relatedness, the raw numbers of co-occurrences between entities cannot straightforwardly be interpreted as giving the strength of the relation between each pair of entities. There is a so-called size effect, as some classes co-occur more often with others for the simple reason that these classes have more occurrences in the first place. In our example, d has more co-occurrences with c than with a or b but c also has more occurrences in total and therefore is more likely to co-occur with any class.

To correct the absolute number of co-occurrences for the size effect a normalization procedure is applied to the data (van Eck & Waltman, 2009)9. Correcting co-occurrence data for the size effect to derive relationships between entities is done through direct similarity measures10. van Eck and Waltman (2009) wrote an extensive review of the most popular direct similarity measures: the cosine, the Jaccard index, the inclusion index, and the association strength. Of these, the last is a probabilistic measure, while the others are set-theoretic measures. The authors show that set-theoretic measures do not properly correct for the size effect and argue in favor of the association strength.

The usability of their formula exceeds the domain of scientometrics. Hidalgo et al. (2007) developed an influential network analysis tool to derive the relatedness between entities on the basis of co-occurrences. Although they use a different probabilistic direct similarity measure than the ones covered by van Eck and Waltman (2009), other authors (e.g., Balland, Rigby, & Boschma, 2015) building on the framework of Hidalgo et al. (2007) do opt for the association strength, as defined by van Eck and Waltman (2009) 11.

Albeit influential, refinements to the work of van Eck and Waltman (2009) are in place. The probabilistic formula should be based on a specific case of combinations without repetition instead of with repetition. Furthermore, the definitions of the inputs for the formula are imprecise. These points will be treated in the following section. It should be noted that the refinements to the measure do not undermine in any way the statement of van Eck and Waltman (2009) that probabilistic measures outperform set-theoretic measures in normalizing co-occurrence data to control for the size effect.

3. REFINEMENT TO THE ASSOCIATION STRENGTH

The objective of the association strength is to estimate the number of expected co-occurrences for each pair, assuming that these are randomly distributed, and compare this to the number of observed co-occurrences to give an indication of the relation between a pair of entities when corrected for the size effect. The challenge therefore is to correctly estimate the number of expected co-occurrences per combination.

As an intuitive example, Matrix 3 gives a co-occurrence matrix C in which three classes (a, b, and c) exist and co-occur exactly once with each other12:

Matrix 3

ClassaClassbClasscClassa011Classb101Classc110

As each class has two observations and two possible other classes to co-occur with, the expected number of co-occurrences is logically 22 = 1 for each combination (a & b, a & c, and b & c).

In this case, the matrix of expected co-occurrences is exactly the same as the matrix of observed co-occurrences given in Matrix 3. Therefore, we observe as many co-occurrences as expected and ObservedExpected should be equal to one for each combination.

For the association strength, van Eck and Waltman (2009) use a simplified formula in the main text but describe Eq. 1 on p. 163613,14:
SOriginalCijSiSjTm=CijSiTSjT+SjTSiTm,ij,
(1)
In which Si and Sj are the number of occurrences of entity i (respectively j) involved in co-occurrences where ij. To calculate Si one can use the row sum of row i of the matrix C when the diagonal is set to zero15. This slightly diverges from the explanation of van Eck and Waltman (2009),16. T is the total number of occurrences and equal to i=1nSi, with n being the total number of entities, and m is the total number of n co-occurrences and therefore equal to i=1nSi2, which is half of T as each co-occurrence involves two occurrences. This definition also diverges from van Eck and Waltman (2009) 17. Cij is the number of observed co-occurrences between i and j.

In essence, the denominator gives that the chance of encountering a co-occurrence between an observation of class i and an observation of class j is equal to the probability of first drawing one of the observations of class i out of the total number of occurrences times the chance of drawing an observation belonging to class j out of the total number of occurrences plus the probability of first drawing j and then i times the total number of co-occurrences.

Calculating this formula for our example C in Matrix 3 would yield Relatedness Matrix R given in Matrix 4:

Matrix 4

ClassaClassbClasscClassa01.51.5Classb1.501.5Classc1.51.50

It is clear that the formula does not provide the intuitive answer of 1 but actually overestimates the relationship by returning that each pair co-occurs more often than could be expected given a random distribution.

The flaw cannot lie in the numerator, which is equal to the number of observed co-occurrences. Therefore the problem lies in the denominator. The formula to calculate the expected number of co-occurrences includes the possibility that when an occurrence of a certain entity is drawn the same occurrence or another occurrence of the same entity (if present) can be drawn in the next draw to complete the co-occurrence. This is known as combinations with repetition. However, as self co-occurrences are nonexistent one knows that one cannot redraw the same occurrence, but also none of the other occurrences of that class.

In the case of our example, the denominator of Eq. 1 yields an expected number of 23 co-occurrences. This is because the formula observes two occurrences for each class and three possible partners to co-occur with, even though there are only two possible partners. Class a can co-occur with class b and class c but not with itself18.

In the case of co-occurrence data in which none of the observations belonging to the previously drawn entity can be drawn in the second draw the correct probabilistic measure would be Eq. 2:
SImprovedCijSiSjTm=CijSiTSjTSi+SjTSiTSim,ij,
(2)

Here, the denominator gives that the chance of encountering a co-occurrence between an observation of class i and an observation of class j is equal to the probability of first drawing one of the observations of class i times the chance of drawing an observation belonging to class j knowing that none of the observations of class i can be drawn plus the chance of first drawing one of the observations of class j times the chance of drawing an observation belonging to class i knowing that any other observations of class j cannot be drawn.

The implications of using Eq. 1 instead of Eq. 2 are that the relatedness between a pair is overestimated when at least one co-occurrence is observed and that the overestimation is larger for certain pairs than others. These implications are demonstrated and further explored in the following parts: first in a theoretic setting, then by running simulations and concluding with the analysis of a real-world example using patent data.

4. THEORETICAL EXPLORATION OF THE OVERESTIMATION

An obvious first notion from observing Eqs. 1 and 2 is that there is no difference in outcome when the number of observed co-occurrences is zero, as the numerator Cij will then be zero.

Furthermore, it can be assumed that Eq. 1 overestimates the relation between two entities when there is at least one co-occurrence. The assumption in the probabilistic measure of Eq. 1 is that the same observation and other observations from the same entity can be drawn again but this is not possible. This enlarges the total pool from which observations can be drawn and therefore decreases the likelihood that a certain co-occurrence can be drawn. This leads to the denominator, which contains the expected number of co-occurrences, in Eq. 1 being smaller than the one in Eq. 2 in all cases, as was the case for the example Matrix 3, where the denominator indicated a co-occurrence probability of 23 for each pair where actually only two options instead of three existed and therefore 22 should have been the answer.

Due to the smaller expected probability, Eq. 1 divides the number of observed co-occurrences over too small a number of expected co-occurrences and therefore the relatedness these two entities is overestimated, when at least one co-occurrence is observed.

That the denominator of Eq. 1 underestimates the expected number of co-occurrences can also be proven analytically. The original probabilistic measure of van Eck and Waltman (2009) in the denominator of Eq. 1 is rewritten and given in Eq. 3, while the improved probabilistic measure used in the denominator of Eq. 2 is rewritten and given in Eq. 4:
ECijOriginalSiSjT=SiSjT,ij,
(3)
ECijImprovedSiSjT=SiSj2TSiSj2TSiTSj,ij,
(4)
Let Dprobability be equal to E(Cij)ImprovedE(Cij)Original. It can be shown that this difference Dprobability is equal to Eq. 5.
DprobabilitySiSjT=SiSjSiT+SjT2SiSj2TTSiTSj,ij,
(5)
For E(Cij)Improved to be larger than E(Cij)OriginalEq. 5 gives that SiT + SjT must be larger than 2SiSj. As Si ≥ 1, Sj ≥ 1, and T = Si + Sj + Sk + … + Sn it is clear that T > Si and T > Sj and therefore SiT + SjT > 2SiSj must hold19.

This means that Dprobability is positive in all circumstances, which indicates that the improved formula predicts in all cases that more co-occurrences can be expected between i and j. This makes sense, as the improved formula excludes the possibility of drawing a combination of i and i, making it more likely to draw a combination between i and j.

Because the number of observed co-occurrences, Cij, is divided over the number of expected co-occurrences, the original Eq. 1 leads to larger results than the improved Eq. 2 in all possible cases, when Cij > 0. This can also be shown mathematically: Let DFormula be equal to SOriginal(Cij, Si, Sj, T) − SImproved(Cij, Si, Sj, T)20. It can be shown that the difference DFormula is equal to Eq. 8 after rewriting Eq. 1 to Eq. 6 and Eq. 2 to Eq. 7.
SOriginalCijSiSjT=TCijSiSj,ij,
(6)
SImprovedCijSiSjT=2TSiTSjCijSiSj2TSiSj,ij,
(7)
DFormulaCijSiSjT=SiT+SjT2SiSjCijSiSj2TSiSj,ij,
(8)
Three important notions can be derived from Eq. 8. First, it is confirmed that when there are no observed co-occurrences (i.e., Cij = 0) the difference is zero. Second, if and only if Cij > 0 then SiSj ≥ 1 and TSi + Sj and therefore (SiT + SjT > 2SiSj). This indicates that Eq. 1 yields larger outcomes than Eq. 2 in all possible cases, with at least one observed co-occurrence, effectively overestimating the relation between entity i and j. Third, for different values of Si, Sj, Cij, and T the difference between Eqs. 1 and 2 will also vary. This means that the difference between the formulas is not proportional for each pair but the relatedness between certain pairs is more strongly overestimated than for other pairs.

To explore the difference due to different values of Si, Sj, Cij, and T the partial derivatives are taken of DFormula with respect to each. Because T is a function of Si, Sj, and all other co-occurrences, ki,jnSk. T is replaced by Si + Sj + L in Eq. 8 in which L = ki,jnSk and its range is equal to or larger than zero.

The partial derivatives δDFormulaδCij, δDFormulaδSi, and δDFormulaδL are respectively given in Eqs. 9, 10, and 1121.
δDFormulaδCij=Si2+Sj2+SiL+SjLSiSjSi+Sj+2L,ij,
(9)
δDFormulaδSi=CijSi2Sj+Si2L+2SiL22SiSjLSj33L2SiL2SjL2Si2SjSi+Sj+2L2,ij,
(10)
δDFormulaδL=CijSiSj2SiSjSi+Sj+2L2,ij,
(11)
Given the domain of each formula, Eq. 9 is always positive, and, when at least one co-occurrence exists, Eq. 10 can be positive or negative depending on the respective inputs and Eq. 11 is always negative.

This last statement suggests that a relationship between two entities will be more overestimated by Eq. 1 when there is a smaller amount of other possibilities to co-occur with.

Despite being informative, partial derivatives give an incomplete picture of the discrepancy between the two formulas, as these give the direction of a function with respect to an infinitesimal increase in one of the variables while keeping the others equal, even though in reality it is impossible to keep the other variables equal, as the inputs are all related to each other. Necessarily Cij consists of Si and Sj, and if not all Si co-occur with Sj then L must at least have enough occurrences to co-occur with the remaining i, and js. In other words, the following logical conditions hold: Cijmin{Si, Sj}; and L ≥ |SiSj|. In the next section theoretical simulations are run in which these conditions can be met.

5. SIMULATIONAL EXPLORATION OF THE OVERESTIMATION

For the theoretical simulations a simple co-occurrence matrix C depicted in Matrix 5 is used. Although it is simple, this matrix allows for some exploration of the numerical difference between Eq. 1 or Eq. 2 for different values of Si, Sj, Cij, and L. In four different simulations, hypothetical and rather extreme situations are simulated to get insight into the effects of increasing the values of each of the variables Si, Sj, Cij, and L, while meeting the conditions Cijmin{Si, Sj}; and L ≥ |SiSj|.

Matrix 5

Classesabcda0111b1011c1101d1110

In the first simulation, Matrix 5 is taken and the number of co-occurrences between c & d is increased by 1 in each step k, ceteris paribus. Matrix 6 gives this simulation.

Matrix 6

Classesabcda0111b1011c1101+kd111+k0

In each step k the resulting relatedness matrix using Eq. 1 is subtracted from the resulting relatedness matrix using Eq. 2 and divided over the value of Eq. 2 to express the difference in percentages. The relatedness values for the pairs a & b and c & d are then plotted for each step. Each of these two changing relationships represent a different scenario:

  • a&b. The changing difference in relatedness for the pair a & b simulates a steady increase in L, keeping Cij = 1 and Si = Sj = 3. This result is depicted in Figure 1.

  • c&d. The changing difference in relatedness between classes c & d simulates a steady increase in Cij but also in Si and Sj, keeping L = 6. To increase Cij beyond the maximum value of Si and Sj, Si and Sj also have to increase. From the partial derivatives it can be derived that an increasing Cij would increase the difference, whereas an increase in Si and Sj can both increase or decrease the difference. The result of the simulation is depicted in Figure 2.

The absolute difference between the calculated relatedness of Eqs. 1 and 2 for the pair a & b is equal to 1/3 across the entire simulation. However, as the number of other co-occurrences L increases, potential co-occurrence candidates increase as well, and therefore the expected number of co-occurrences for a & b decreases. As a result, relatedness values are higher as L increases and the relative difference decreases, as can be seen in Figure 1.

Figure 1.

The difference in relatedness between the original formula and the improved formula for class a & b when L increases.

Figure 1.

The difference in relatedness between the original formula and the improved formula for class a & b when L increases.

For pair c & d, L remains equal to 6 but Ccd, Sc, and Sd increase. Figure 2 depicts how the difference in the estimated relatedness increases asymptotically, converging from 33.3% to the value of 100%. As the ObservedExpected should be close to 1 when two entities are close to having 100% of the occurrences in the sample, but the values of the original Eq. 1 converge to 2 the difference is close to 100% of the correct value.

Figure 2.

The difference in relatedness between the original formula and the improved formula for class c & d when Ccd, Sc, and Sd increase.

Figure 2.

The difference in relatedness between the original formula and the improved formula for class c & d when Ccd, Sc, and Sd increase.

To simulate an increase in Cij while keeping Si, Sj, and L equal, ceteris paribus, another simulation is needed: Matrix 1 is altered by replacing the number of co-occurrences between entities a & b and c & d by a large amount of co-occurrences x.

Then in each step k of the simulation a co-occurrence is subtracted from this amount x and added to the co-occurrences between entities a & d and b & c: See Matrix 6. This keeps Si, Sj, and L equal but increases Cij for the relatedness between a & d. Note that the result is insensitive to the exact value of x, as the resulting change in the denominator and numerator cancel each other out.

Matrix 7

Classesabcda0xk11+kbxk01+k1c11+k0xkd1+k1xk0

The result is a stable overestimation of 33.3% for all values of k. When a & d co-occur more often but the total number of co-occurrences in the sample stays the same, the relatedness between a & d naturally increases. Nonetheless, the increase in relatedness is proportional for the two formulas and therefore the difference remains 33.3%.

Last, an increase in Si and Sj while keeping Cij equal is simulated. The simulation is very similar to the first simulation except that in addition to increasing the co-occurrences between c & d also those between b & c are increased in each step k: See Matrix 4. As a result, Sb and Sc increase while Cbd is kept at 1. L necessarily increases as well in the form of Sc to match the added co-occurrences of Sb and Sd.

Matrix 8

Classesabcda0111b101+k1c11+k01+kd111+k0

Once again the percentage difference between calculating the level of relatedness for the pair b & d using Eqs. 1 and 2 is stable at 33.3% for all values k. This time the relatedness between b & d decreases as k increases because their total number of occurrences Sb and Sd increase but their number of co-occurrences remains 1.

The simulations in this section show that the difference can range between close to 100% and close to 0. In real-world applications of co-occurrence data the bias introduced by using Eq. 1 instead of Eq. 2 will be somewhere in between the extreme scenarios simulated here, in which each respective value in the relatedness matrix will be closer to a specific scenario than others.

6. REAL-WORLD DATA-BASED EXPLORATION OF THE OVERESTIMATION

The theoretical and simulational explorations demonstrate that Eq. 1 overestimates the relatedness between entities compared to Eq. 2 in a way that disproportionately affects certain pairs more than other pairs. However, the question remains how close these examples are to real-world applications.

Therefore, the outcomes of Eqs. 1 and 2 are compared using United States Patent and Trademark Office (USPTO) technology class data: See Hall, Jaffe, and Trajtenberg (2001) and USPTO, from utility patents in periods of 5 years from 1855 to 201422.

In the occurrence matrix O of each time period the rows indicate patent numbers and the columns technology classes, like the example in Matrix 1. By multiplying the transpose of O by O itself a technology classes by technology classes co-occurrence matrix C is obtained. As before, the diagonal of C is set to zero and Si can then be calculated as the column sum of column i or the row sum of row i23. Next Eqs. 1 and 2 are calculated using the C of each time period and the results are compared in Table 1.

Table 1.

Patent comparison results

 1855–91860–41865–91870–41875–91880–41885–91890–4
Number of technology classes 327 335 343 356 361 372 379 385 
Number of related pairs (Original formula) 5,154 4,902 7,910 8,954 10,100 12,396 13,438 13,484 
Number of related pairs (Improved formula) 5,150 4,898 7,892 8,934 10,080 12,370 13,398 13,464 
Difference 18 20 20 26 40 20 
Difference (%) 0.07 0.08 0.22 0.22 0.19 0.21 0.29 0.14 
Largest difference in value 0.827 0.837 0.788 0.786 0.822 0.662 0.593 0.63 
Largest difference (%) in value 2.643 2.177 2.009 1.961 2.258 2.333 2.425 2.36 
Smallest difference in value 0.00599 0.00505 0.00234 0.00169 0.0011 0.00107 0.00084 0.00082 
Smallest difference (%) in value 0.0294 0.0268 0.01 0.0075 0.0085 0.0037 0.004 0.0032 
 1895–91900–41905–91910–41915–91920–41925–91930–4
Number of technology classes 385 387 390 394 403 404 405 415 
Number of related pairs (Original formula) 14,196 15,866 16,372 16,742 17,784 18,036 19,560 21,432 
Number of related pairs (Improved formula) 14,160 15,842 16,338 16,694 17,754 17,990 19,528 21,396 
Difference 36 24 34 48 30 46 32 36 
Difference (%) 0.25 0.15 0.20 0.28 0.16 0.25 0.16 0.16 
Largest difference in value 0.625 0.515 0.586 0.666 0.645 0.753 0.711 0.677 
Largest difference (%) in value 2.568 2.341 2.303 2.441 2.536 2.173 1.933 1.872 
Smallest difference in value 0.00063 0.00056 0.00042 0.00051 0.00038 0.00039 0.00023 0.00018 
Smallest difference (%) in value 0.0026 0.0054 0.0055 0.0071 0.0052 0.0023 0.0036 0.0071 
 1935–91940–41945–91950–41955–91960–41965–91970–4
Number of technology classes 414 417 413 423 427 430 432 434 
Number of related pairs (Original formula) 22,852 23,430 23,336 25,104 24,422 25,326 25,932 25,590 
Number of related pairs (Improved formula) 22,814 23,388 23,280 25,060 24,360 25,280 25,902 25,544 
Difference 38 42 56 44 62 46 30 46 
Difference (%) 0.16 0.17 0.24 0.17 0.25 0.18 0.11 0.18 
Largest difference in value 0.557 0.56 0.525 0.492 0.557 0.529 0.579 0.661 
Largest difference (%) in value 1.641 1.76 1.772 1.726 1.51 1.561 1.602 1.892 
Smallest difference in value 0.00015 0.00015 0.00022 0.00014 0.00014 0.00011 0.00009 0.00008 
Smallest difference (%) in value 0.003 0.0034 0.0063 0.0019 0.0029 0.0018 0.0015 0.0006 
 1975–91980–41985–91990–41995–92000–42005–92010–4
Number of technology classes 436 435 435 435 431 437 436 438 
Number of related pairs (Original formula) 25,350 25,012 24,712 23,982 24,120 24,422 24,356 26,382 
Number of related pairs (Improved formula) 25,324 24,980 24,676 23,928 24,084 24,388 24,310 26,348 
Difference 26 32 36 54 36 34 46 34 
Difference (%) 0.10 0.12 0.14 0.22 0.14 0.13 0.18 0.12 
Largest difference in value 0.684 0.694 0.69 0.524 0.501 0.581 0.592 0.64 
Largest difference (%) in value 2.29 2.52 2.192 2.293 2.404 3.234 3.176 2.834 
Smallest difference in value 0.00008 0.00008 0.00006 0.00005 0.00005 0.00004 0.00003 0.00002 
Smallest difference (%) in value 0.0018 0.0033 0.004 0.0028 0.0033 0.0036 0.0013 0.0012 
 1855–91860–41865–91870–41875–91880–41885–91890–4
Number of technology classes 327 335 343 356 361 372 379 385 
Number of related pairs (Original formula) 5,154 4,902 7,910 8,954 10,100 12,396 13,438 13,484 
Number of related pairs (Improved formula) 5,150 4,898 7,892 8,934 10,080 12,370 13,398 13,464 
Difference 18 20 20 26 40 20 
Difference (%) 0.07 0.08 0.22 0.22 0.19 0.21 0.29 0.14 
Largest difference in value 0.827 0.837 0.788 0.786 0.822 0.662 0.593 0.63 
Largest difference (%) in value 2.643 2.177 2.009 1.961 2.258 2.333 2.425 2.36 
Smallest difference in value 0.00599 0.00505 0.00234 0.00169 0.0011 0.00107 0.00084 0.00082 
Smallest difference (%) in value 0.0294 0.0268 0.01 0.0075 0.0085 0.0037 0.004 0.0032 
 1895–91900–41905–91910–41915–91920–41925–91930–4
Number of technology classes 385 387 390 394 403 404 405 415 
Number of related pairs (Original formula) 14,196 15,866 16,372 16,742 17,784 18,036 19,560 21,432 
Number of related pairs (Improved formula) 14,160 15,842 16,338 16,694 17,754 17,990 19,528 21,396 
Difference 36 24 34 48 30 46 32 36 
Difference (%) 0.25 0.15 0.20 0.28 0.16 0.25 0.16 0.16 
Largest difference in value 0.625 0.515 0.586 0.666 0.645 0.753 0.711 0.677 
Largest difference (%) in value 2.568 2.341 2.303 2.441 2.536 2.173 1.933 1.872 
Smallest difference in value 0.00063 0.00056 0.00042 0.00051 0.00038 0.00039 0.00023 0.00018 
Smallest difference (%) in value 0.0026 0.0054 0.0055 0.0071 0.0052 0.0023 0.0036 0.0071 
 1935–91940–41945–91950–41955–91960–41965–91970–4
Number of technology classes 414 417 413 423 427 430 432 434 
Number of related pairs (Original formula) 22,852 23,430 23,336 25,104 24,422 25,326 25,932 25,590 
Number of related pairs (Improved formula) 22,814 23,388 23,280 25,060 24,360 25,280 25,902 25,544 
Difference 38 42 56 44 62 46 30 46 
Difference (%) 0.16 0.17 0.24 0.17 0.25 0.18 0.11 0.18 
Largest difference in value 0.557 0.56 0.525 0.492 0.557 0.529 0.579 0.661 
Largest difference (%) in value 1.641 1.76 1.772 1.726 1.51 1.561 1.602 1.892 
Smallest difference in value 0.00015 0.00015 0.00022 0.00014 0.00014 0.00011 0.00009 0.00008 
Smallest difference (%) in value 0.003 0.0034 0.0063 0.0019 0.0029 0.0018 0.0015 0.0006 
 1975–91980–41985–91990–41995–92000–42005–92010–4
Number of technology classes 436 435 435 435 431 437 436 438 
Number of related pairs (Original formula) 25,350 25,012 24,712 23,982 24,120 24,422 24,356 26,382 
Number of related pairs (Improved formula) 25,324 24,980 24,676 23,928 24,084 24,388 24,310 26,348 
Difference 26 32 36 54 36 34 46 34 
Difference (%) 0.10 0.12 0.14 0.22 0.14 0.13 0.18 0.12 
Largest difference in value 0.684 0.694 0.69 0.524 0.501 0.581 0.592 0.64 
Largest difference (%) in value 2.29 2.52 2.192 2.293 2.404 3.234 3.176 2.834 
Smallest difference in value 0.00008 0.00008 0.00006 0.00005 0.00005 0.00004 0.00003 0.00002 
Smallest difference (%) in value 0.0018 0.0033 0.004 0.0028 0.0033 0.0036 0.0013 0.0012 

Notes: A pair is seen as related when the respective formula returns a value of 1 or higher for a certain pair. The statistics expressed in percentages are taken with respect to the value returned by the improved Eq. 2.

Table 1 gives a number of statistics for each time period mentioned in the respective header. The first row gives the number of different technology classes (n) referred to on the patents. This number is equal to the number of columns/rows in C. The second line gives the number of pairs that have a value higher than 1 according to Eq. 1 by van Eck and Waltman (2009); these relatedness pairs have more or just as many observed co-occurrences as expected and are therefore seen as related in research within this domain (see, for example, Balland et al., 2015). The third line gives the same statistic but employs the improved Eq. 2. On line four the difference between the number of related pairs according to each formula is given24. Difference (%) expresses this difference as a percentage of the number of related pairs according to the improved Eq. 2.

By focusing on these first five statistics it can be seen that in 1855–1859 patents made references to 327 different technology classes and that according to Eq. 1 5,154 pairs of technology classes can be seen as related, whereas Eq. 2 identifies 5,150 related pairs. As a result, Eq. 1 identifies four pairs or 45150 × 100 = 0.07% more as related than Eq. 2.

In later time periods the differences increase both in absolute terms and in relative terms with a maximum in relative terms of 0.29% in 1885–1889 and a maximum in absolute terms with 62 pairs wrongly seen as related in 1955–1959.

In addition to the overestimation, another problem of using Eq. 1 instead of Eq. 2 is that the relatedness between some pairs is more overestimated than between other pairs. The last four statistics explore this disproportionality. The largest difference in value gives the largest difference in the relatedness value of a single pair between Eqs. 1 and 2, and its percentage counterpart gives the largest overestimation relative to the value given by Eq. 2. In relative terms the highest over estimation is 3.23% and occurs in 2000–2004, and this percentage is way below some of the extreme scenarios simulated in Section 5. The largest absolute difference is 0.837 in 1860–1864.

The last two statistics are similar but give the smallest difference when Cij > 025. When at least one co-occurrence exists between a pair its relation is overestimated, as already shown mathematically in Section 4. The values are close to zero both in absolute terms and in relative terms and therefore in strong contrast to the highest values, showing that some pairs get more overestimated than others.

The results also show that there is not necessarily a direct connection between the number of technology classes and the number of related pairs or the overestimation. In 2000–2004, there is the second highest number of different technology classes, and the number of related pairs is lower than in 1950–1954 when fewer technology classes were in use.

When comparing these specific time periods, 2000–2004 turns out to have a much more concentrated co-occurrence matrix C than the one in 1950–1954. In 2000–2004 each row or column i contains a few pairs with a lot of observations whereas others have relatively few observations. This contrasts with the more even spread of observations across C in 1950–1954. The average Gini coefficient per row of C in 2000–2004 is 0.936 versus 0.909 in 1950–1954.

Very much like the simulation based on Matrix 7, where Si and Sj were increased while keeping Cij equal, the pairs with little co-occurrence are less overestimated when there are more occurrences of the same technology class with other classes, as is more the case in 2000–2004. The pairs with relatively high numbers of co-occurrences have a larger share of the sample in 2000–2004 compared to 1950–1954, as in Matrix 6, where Cij is increased while Si and Sj are kept equal; these pairs are more overestimated in 2000–2004. The pairs with relatively many co-occurrences are likely to pass the threshold of 1 using either formula, and the stronger overestimation for these pairs in 2000–2004 does not lead to much change with respect to passing this threshold. This is not the case for the pairs with relatively fewer co-occurrences, which are less overestimated in 2000–2004 than in 1950–1954. Therefore, in 2000–2004 these are less likely to pass the threshold irrespective of whether Eq. 1 or Eq. 2 is used, and in 1950–1954 these pairs are more likely to pass the threshold using Eq. 1 but not when using Eq. 2. As a result, 2000–2004 has larger overestimations of individual relatedness values but fewer pairs that are wrongly identified as related.

The comparison shows that using Eq. 1 instead of Eq. 2 in research can lead to nonnegligible differences and that some pairs and matrices are affected disproportionately. Note that with an incorrect specification of Si, Sj, and mEq. 1 becomes even more inaccurate: See Section 4. It is unlikely that papers employing Eq. 1 instead of Eq. 2 would have reached fundamentally different conclusions, but a risk is more present in some cases than others. It is recommended to use Eq. 2 in future research.

7. CONCLUSION

Co-occurrence data is commonly used in various domains. Researchers generally apply normalization measures to correct for the size effect. To this end, van Eck and Waltman (2009) make a convincing case to use a probability-based measure known as the association strength, in which the number of observed co-occurences is divided over the number of expected co-occurrences, assuming that observations are randomly distributed over co-occurences.

However, the probability formula to calculate the expected number of co-occurrences is not suited for the co-occurrence analysis it is recommended for, which is when self-co-occurrences are nonexistent or irrelevant26. The formula assumes combinations with repetition, meaning that an observation from an entity can be drawn again after being picked in the first draw, even though neither this occurrence nor any other occurrence belonging to the same entity can be drawn in this line of work.

This paper introduces a formula that is based on, but not equal to, combinations without repetition in which the probability of drawing entity i and j together is calculated as the probability of drawing i first and then j, knowing that none of the observations pertaining to i can be drawn plus the probability of drawing j and then i, knowing that none of the observations pertaining to j can be drawn. This formula gives the correct results, as was demonstrated in an intuitive example.

Furthermore, it is shown that the original formula overestimates the relatedness between a pair of entities compared to the improved formula introduced here when there is at least one observed co-occurrence, and that the overestimation is not proportional across pairs. Simulations show that the overestimation of the relatedness can range between virtually 0% and almost 100% of the correct value given by the improved formula. In a real-world example, a number of patent samples showed that the overestimation of individual values was between virtually 0% and 3.234%, and the difference in the number of pairs that can be seen as related can be 0.29% more than the number of pairs identified as related by the improved formula.

This paper shows that the formula presented here is better equipped for the analysis of co-occurrence data. The formula, following all recommendations for inputs and treatment, is available in the EconGeo package for R maintained by Balland (2016).

ACKNOWLEDGMENTS

I thank Ludo Waltman, Nees Jan van Eck, Pierre-Alexandre Balland, Ron Boschma, and two anonymous referees for useful comments on this paper. All errors remain my own.

COMPETING INTERESTS

The author has no competing interests.

FUNDING INFORMATION

This work has benefited from grant 438-13-406 from JPI Urban Europe.

DATA AVAILABILITY

The data used in Section 6 comes from Hall et al. (2001) and the USPTO and is freely available from the website of the USPTO.

Notes

1

Note that it depends on the goal of the research if it is necessary to correct for the size effect or that absolute counts are more relevant. In the research cited here and in van Eck and Waltman (2009), normalization is assumed to be necessary. The exact definitions of occurrences, co-occurrences, and the size effect are given in Section 3.

2

A value of one indicates that exactly the same number of co-occurrences are observed as are expected. A value above one or below one indicates, respectively, a stronger relation or a weaker relation between the two entities.

3

This holds for the work referred to in this paper and those by van Eck and Waltman (2009).

4

In this paper, the suggestion is made to set them to zero (see Section 2), which is also often used (Ahlgren, Jarneving & Rousseau, 2003).

5

This type of matrix, in which two sets of vertices, here places and entities, are connected by the co-occurrences in such a way that each link is between one entity and one place, is also known as a bipartite matrix in graph theory (Latapy, Magnien, & Vecchio, 2008).

6

There are, for example, occurrence matrices of scientific publications by authors (e.g., Leydesdorff & Vaughan, 2006) or by research institutions (e.g., Hoekman, Frenken, & Tijssen, 2010); countries by industries (e.g., Hidalgo et al., 2007); streams by fish species (e.g., Peres-Neto, 2004); and patent documents by technology classes (e.g., Boschma et al., 2015).

7

If the rows of O indicate the entities and the columns indicate the places where they co-occur, then it is the other way around, and O should be multiplied by its transpose.

8

Ahlgren et al. (2003) also mention the option of setting the diagonal equal to the number of times an entity occurs at least twice in a place. This option is unsuitable for probabilistic similarity measures, such as the association strength, because the number of times an entity occurs at least twice does not entail a co-occurrence between i and j. Therefore, when estimating the probability of a co-occurrence between i and j one cannot draw the observations on the diagonal even though these are added to the total, and therefore the pool of observations from which one can draw. This becomes clearer when discussing the formula in Section 3.

9

In some cases, more normalization measures are deemed necessary. For example, Neffke, Henning, and Boschma (2011) who look at the co-occurrence of products in the production process of the same plant, also correct for the profitability of the respective products.

10

Another option to derive similarities or relationships between entities is by comparing the co-occurrence profiles of the entities, which are known as indirect similarity measures (van Eck & Waltman, 2009).

11

Hidalgo et al. (2007) look into the co-occurrence of specializations in exporting industries in a country. Their formula consists of taking the smallest value of the conditional probability of effectively exporting product j knowing that a country effectively exports i and the conditional probability of effectively exporting product i knowing that a country effectively exports j. This does not properly correct for the size effect because each conditional probability corrects for the size of only one of the two, the former of i and the latter of j; by picking the smallest of the two conditional probabilities, the other size effect still remains. Furthermore, the probability if a country meets the condition of effectively exporting product j or i is neglected by taking the conditional probabilities. These reasons make it understandable that other authors following the line of Hidalgo et al. (2007) have opted for the association strength of van Eck and Waltman (2009).

12

This matrix C would result from our example O in Matrix 1 if one were to remove class d and its observations.

13

I argue that it is more advantageous to use the full formula, which entails exactly dividing the number of observed co-occurrences over the number of expected co-occurrences, as it gives a clear threshold of one when Observed = Expected. As such, values below one indicate that fewer co-occurrences are observed than could be expected given a random distribution, whereas values above indicate the opposite. This threshold holds in all cases, even when matrices with different numbers of occurrences are compared. In contrast, the simplified formula would have a different value indicating that the number of observed co-occurrences equals the expected number, depending on the matrices, even though it is proportional to the more detailed formula by a factor of 2m.

14

This formula is also presented in rewritten form in Eq. 1 in Waltman, van Eck, and Noyons (2010).

15

Taking the column sum of column i gives the same value as the row sum of row i.

16

van Eck and Waltman (2009, p. 1636) state that for Si both the number of occurrences of entity i can be used or the number of co-occurrences in which i is involved. However, it is important to emphasize that single occurrences, as in Patent 1 of the example O in Matrix 1, should be ignored, as these do not lead to co-occurrences. This also holds for self co-occurrences of i with i, as both of these cannot be part of Cij where ij. Setting the diagonal to zero resolves both these issues. This is also the reason that setting the diagonal equal to the number of times an entity occurs at least twice in a place, as suggested by Ahlgren et al. (2003), is unsuitable for this probabilistic measure.

17

van Eck and Waltman (2009, p. 1648) state that m should be equal to “the number of documents.” However, this only holds when the number of documents is equal to the number of co-occurrences. In the example O in Matrix 1 patent 1 is one document but only refers to one class, so it does not involve any co-occurrences and is therefore not equal to one co-occurrence. Patent 3, on the other hand, is also a single document but refers to all classes a to d and therefore leads to six unique co-occurrences (a&b, a&c, a&d, b&c, b&d, c&d). All together the example consists of three documents and seven unique co-occurrences. As a result, in this case using the number of documents for m would underestimate the expected number of co-occurrences, as the probability of encountering a co-occurrence is multiplied by too small a number of co-occurrences than are actually possible. This explanation is the same as in Waltman et al. (2010). From this follows that the size effect is the result of the fact that some entities are involved in more co-occurrences than others, which means more observations and therefore an increased likelihood to co-occur with any other entity. This means that the raw probabilities of co-occurrence cannot be compared straight away and a normalization measure is needed, such as the one introduced in this paper.

18

To be exact, the denominator of Eq. 1 would be equal to (2626 + 2626)3 for each pair outside of the diagonal in the matrix of this example.

19

If entities can partially occur in a place then the values for Si and Sj can be below one, but in any case not below or equal to zero, and therefore same statements hold.

20

Note that the order of the original formula and the improved formula has been altered compared to the previous calculation of the difference of the respective probabilistic measures.

21

The partial derivatives δDFormulaδSi and δDFormulaδSj are very similar in the sense that one can interchange the Si and Sj to obtain the same formula; therefore δDFormulaδSj is not shown.

22

A period of 5 years is also used by Boschma et al. (2015).

23

Note that the relatedness function in the EconGeo package for R (Balland, 2016) sets the diagonal of the input co-occurrence matrix to zero automatically.

24

Note that there are no pairs identified as related by Eq. 2 that are identified as unrelated by Eq. 1, as Eq. 1 > Eq. 2, when Cij > 0. See also Section 4.

25

When Cij = 0 both formulas return 0 and the difference is therefore also zero and obviously the smallest.

26

An interesting avenue for future research may be to more clearly determine in which situations self-co-occurrences can be disregarded or not.

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Author notes

Handling Editor: Staša Milojević

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